Dynamical Systems Analysis for Systems of Spiking Neurons

Dynamical Systems Analysis for Systems of Spiking Neurons

Models: Leaky Integrate and Fire Model Cd. V/dt= -V/R+Isyn • Resting Potential VRest assumed to be 0. • CR = Membrane time constant (20 msec for excitatory neurons, 10 msec for inhibitory neurons. ) • Spike generated when V reaches VThreshold • Voltage reset to VReset after spike (not the same as VRest) • Synaptic Current Isyn assumed to be either delta function or alpha function.

Models: Spike-Response Model Observation: The L-IF-model is linear Cd. V 1/dt= -V 1/R+I 1 syn Cd. V 2/dt= -V 2/R+I 2 syn Cd(V 1+V 2)/dt= -(V 1+V 2)/R+I 1 syn+I 2 syn Why not simply take the individual effect of each spike and add them all up? Result: The Spike response model. V(t)=effect of previously generated spikes by neuron+ sum over all effects generated by spikes that have arrived at synapses

Background: The Cortical Neuron • Synapse • Dendrites (Input) • Cell Body • Axon (Output) Output Input Threshold • Absolute Refractory Period • Exponential Decay of effect of a spike on membrane potential Time

Background: Target System Neocortical Column: ~ 1 mm 2 of the cortex Output Recurrent network ~100, 000 neurons ~10, 000 synapses per neuron ~80% excitatory ~20% inhibitory Recurrent System Input

Background: The Neocortex (Healthy adult human male subject) Source: Dr. Krishna Nayak, SCRI, FSU

Background: The Neocortex (Area V 1 of Macaque Monkey) Source: Dr. Wyeth Bair, CNS, NYU

Background: Dynamical Systems Analysis Phase Space • Set of all legal states Dynamics • Velocity Field • Flows • Mapping Local & Global properties • Sensitivity to initial conditions • Fixed points and periodic orbits

Content: • Model • A neuron • System of Neurons: Phase Space & Velocity Field • Simulation Experiments • Neocortical Column • Qualitative Characteristics: EEG power spectrum & ISI frequency distribution • Formal Analysis • Local Analysis: Sensitivity to Initial Conditions • Conclusions

Model: Single Neuron t=0 t=0 Potential Function Each spike represented as: How long since it departed from soma. Time

Model: Single Neuron: Potential function

Model: System of Neurons • Dynamics • Birth of a spike • Death of a spike • Point in the Phase-Space • Configuration of spikes

Model: Single Neuron: Phase-Space

Model: Single Neuron: Phase-Space Theorem: Theorem Phase-Space can be defined formally Phase-Space for Total Number of Spikes Assigned = 1, 2, & 3.

Model: Single Neuron: Structure of Phase-Space • Phase-Space for n=3 • 1, 2 dead spikes.

Model: System of Neurons: Velocity Field

Simulations: Neocortical Column: Setup • 1000 neurons each connected randomly to 100 neurons. • 80% randomly chosen to be excitatory, rest inhibitory. • Basic Spike-response model. • Total number of active spikes in the system ►EEG / LFP recordings • Spike Activity of randomly chosen neurons ►Real spike train recordings • 5 models: Successively enhanced physiological accuracy • Simplest model • Identical EPSPs and IPSPs, IPSP 6 times stronger • Most complex model • Synapses: Excitatory (50% AMPA, NMDA), Inhibitory (50% GABAA, GABAB) • Realistic distribution of synapses on soma and dendrites • Synaptic response as reported in (Bernander Douglas & Koch 1992)

Simulations: Neocortical Column: Classes of Activity Number of active spikes: spikes Seizure-like & Normal Operational Conditions

Simulations: Neocortical Column: Chaotic Activity T=0 T=1000 msec Normal Operational Conditions (20 Hz): Hz) Subset (200 neurons) of 1000 neurons for 1 second.

Simulations: Neocortical Column: Total Activity Normalized time series: series Total number of active spikes & Power Spectrum

Simulations: Neocortical Column: Spike Trains Representative spike trains: trains Inter-spike Intervals & Frequency Distributions

Simulations: Neocortical Column: Propensity for Chaos ISI’s of representative neurons: neurons 3 systems; 70%, 80%, 90% synapses driven by pacemaker

Simulations: Neocortical Column: Sensitive Dependence on Initial Conditions T=0 T=400 msec Spike activity of 2 Systems: Systems Identical Systems, subset (200) of 1000 neurons, Identical Initial State except for 1 spike perturbed by 1 msec.

Analysis: Local Analysis • Are trajectories sensitive to initial conditions? conditions • If there are fixed points or periodic orbits, orbits are they stable? stable

Analysis: Setup: Riemannian Metric

Analysis: Setup: Riemannian Metric • Discrete Dynamical System • Event ►Event…. • Event: birth/death of spike

Analysis: Measure Analysis Death of a Spike PI Birth of a Spike

Analysis: Perturbation Analysis

Analysis: Perturbation Analysis

Analysis: Local Cross-Section Analysis AT B Birth C Death If then sensitive to initial conditions. If then insensitive to initial conditions.

Analysis: Local Cross-Section Analysis

Analysis: Local Cross-Section Analysis: Prediction

Analysis: Local Cross-Section Analysis: Prediction Seizure Normal Spike rate >1 =1 <1 Neocortical Column =1

Analysis: Discussion • Existence of time average • Systems without Input and with Stationary Input Transformation invariant (Stationary) Stationary Probability measure exists. System has Ergodic properties. • Systems with Transient Inputs ? • Information Coding (Computational State vs. Physical State) • Attractor-equivalent of class of trajectories.